A global multiplicity result for N-Laplacian with critical nonlinearity of concave-convex type

Abstract

Let Ω ⊂ R N , N 2, be a bounded domain. We consider the following quasilinear problem depending on a real parameter λ > 0:where f (t) is a nonlinearity that grows like e t N/N−1 as t → ∞ and behaves like t α , for some α ∈ (0, N − 1), as t → 0 + . More precisely, we require f to satisfy assumptions (A1)-(A5) in Section 1. With these assumptions we show the existence of Λ > 0 such that (P λ ) admits at least two solutions for all λ ∈ (0, Λ), one solution for λ = Λ and no solution for all λ > Λ. We also study the problem (P λ ) posed on the ball B 1 (0) ⊂ R N and show that the assumptions (A1)-(A5) are sharp for obtaining global multiplicity. We use a combination of monotonicity and variational methods to show multiplicity on general domains and asymptotic analysis of ODEs for the case of the ball.

Isolated singularities for the exponential type semilinear elliptic equation in ℝ²

“…Therefore, the curve C bends to the left at λ = Λ. Appealing to the uniqueness and multiplicity result in [11] (see Theorems 1.2,1.3 and Proposition 8.3), we complete the proof of (3). If f is of the form (f 1), from property (4) and the global bifurcation theory of Rabinowitz (see [14]) we see that there exists (λ n , u n ) ∈ C and λ * > 0 such that λ n → λ * and u n (0) → ∞ (since C cannot "cross" the minimal solutions branch which is locally unique).…”

Multiple positive solutions of singular and critical elliptic problem in $${\mathbb{R}^2}$$ with discontinuous nonlinearities

“…The study of Palais-Smale level is also delicate due to the effect of discontinuous nature of the sublinear term. We use sequence of Moser functions with variable support as in [12] to obtain the Palais-Smale sequence below the critical level. Here we would like to mention that the results obtained are new even for the case β = 0.…”

“…Existence results for semilinear equation with continuous and exponential nonlinearity motivated from Moser-Trudinger inequality has been extensively studied starting from [1][2][3]10]. The combined effects of concave and convex nonlinearities are studied in the beautiful work of Ambrosetti et al [4] for critical exponent problems and these results are discussed for the exponential nonlinearities in [12].…”

A note on a dirichlet problem with concave-convex nonlinearity

“…In [7] different than our techniques, such as elliptic estimates, variational methods or comparison principles are used. The variational approach of [7] is based on the application of the mountain pass geometry.…”

“…In [7] different than our techniques, such as elliptic estimates, variational methods or comparison principles are used. The variational approach of [7] is based on the application of the mountain pass geometry. The abstract approach, although in a sublinear case and with different type of nonlinearity is investigated in [2] but this applies only for such differential operators that can represent a duality mapping between suitable spaces.…”